Improving Sustainability Index of Grey Cast Iron Finish Cutting Through High-Speed Dry Turning and Cutting Parameters Optimization Using Taguchi-Based Bayesian Method

Abstract

Grey cast iron (GCI) is the most common metal casting product in the world. Therefore, achieving sustainable production for this particular material will provide a significant impact on the sustainability issue of metal industries in general. This paper attempts to enhance the sustainability of grey cast iron turning through a high speed, dry turning process. The effect of cutting parameters on three assessments of product sustainability index (PSI) called energy consumption (EC), tool cost (TC), and surface roughness (SR) was investigated using the Taguchi method. The analysis of the signal to noise (S/N) ratio shows that the minimum depth of cut (0.1 mm) gives the optimal performance for all three assessments. Meanwhile, the cutting speed and feed rate reveal a conflict in obtaining the best performance. To help with making a decision, a Taguchi-based Bayesian optimization method was proposed and the capability of the method on suppressing the number of samples to obtain the optimum cutting parameters was demonstrated. Furthermore, the reasons for the optimum cutting condition were also presented.

Appendix: Bayesian Optimization Procedure

The prediction $\widehat{y}\left({\varvec{x}}\right)$, which corresponds to the cost function for a vector variable ${\varvec{x}}={\left[{x}_{1},{x}_{2},\ldots ,{x}_{m}\right]}^{T}$ composed of operation conditions in turning is given as follows:

$$\widehat{y}\left({\varvec{x}}\right)=\mu +{{\varvec{\psi}}}^{T}{\Psi }^{-1}\left({\varvec{y}}-1\mu \right),$$

(10)

where ${\varvec{y}}={\left[{y}_{1},{y}_{2},\ldots ,{y}_{n}\right]}^{T}$ corresponds to a vector of the cost function that corresponds to $n$ different experiments, and 1 represents a unit vector. The subscript $m$ indicates the number of operational parameters considered for constructing the model. The covariance matrix is defined as:

$$\Psi =\left[\begin{array}{ccc}cor\left[Y\left({{\varvec{x}}}^{\left(1\right)}\right),Y\left({{\varvec{x}}}^{\left(1\right)}\right)\right]& \cdots & cor\left[Y\left({{\varvec{x}}}^{\left(1\right)}\right),Y\left({{\varvec{x}}}^{\left(n\right)}\right)\right]\\ \vdots & \ddots & \vdots \\ cor\left[Y\left({{\varvec{x}}}^{\left(n\right)}\right),Y\left({{\varvec{x}}}^{\left(1\right)}\right)\right]& \cdots & cor\left[Y\left({{\varvec{x}}}^{\left(n\right)}\right),Y\left({{\varvec{x}}}^{\left(n\right)}\right)\right]\end{array}\right],$$

(11)

where the correlation function is defined as follows:

$$cor\left[Y\left({{\varvec{x}}}^{\left(k\right)}\right),Y\left({{\varvec{x}}}^{\left(l\right)}\right)\right]=\mathrm{exp}\left[-\sum_{j=1}^{m}{\theta }_{j}{|{x}_{j}^{\left(k\right)}-{x}_{j}^{\left(l\right)}|}^{2}\right].$$

(12)

The vector ${\varvec{\psi}}$ in Eq. (10) is the correlation of the sampled parameter sets and a parameter set ${\varvec{x}}$ to be predicted as follows:

$${\varvec{\psi}}=\left[\begin{array}{c}cor[Y({{\varvec{x}}}^{(1)}),Y({\varvec{x}})]\\ \vdots \\ cor[Y({{\varvec{x}}}^{(n)}),Y({\varvec{x}})]\end{array}\right].$$

(13)

The average component $\mu$ in Eq. (10) is defined as $\mu ={(1}^{T}{\Psi }^{-1}{\varvec{y}})/({1}^{T}{\Psi }^{-1}1)$ and the variance ${\sigma }^{2}$ is defined as follows:

$${\sigma }^{2}=\frac{{\left({\varvec{y}}-1\mu \right)}^{T}{\Psi }^{-1}({\varvec{y}}-1\mu )}{n}.$$

(14)

The hyperparameter ${\varvec{\theta}}={\left[{\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{m}\right]}^{T}$ used in Eq. (12) is obtained by maximizing a ln-likelihood function given as:

$$Ln\left({\varvec{\theta}}\right)\approx -\frac{n}{2}\mathrm{ln}\left({\sigma }^{2}\right)-\frac{n}{2}\mathrm{ln}\left|\Psi \right|.$$

(15)

The maximization of the ln-likelihood function is performed using a genetic algorithm to find the hyperparameter ${\varvec{\theta}}$ that best represents the sampled data sets.

Once the surrogate model of a cost function is constructed based on the above procedures, it is possible to minimize/maximize a function value as well as to evaluate indices for improving the surrogate model itself. Both purposes can be achieved simultaneously using expected improvement realizing so-called Bayesian optimization. The expected improvement can be calculated by:

$$EI\left({\varvec{x}}\right)=\left[{y}_{\mathrm{min}}-\widehat{y}\left({\varvec{x}}\right)\right]\left[\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left[\frac{{y}_{\mathrm{min}}-\widehat{y}\left({\varvec{x}}\right)}{\sqrt{2}\widehat{s}}\right]\right]+\widehat{s}\frac{1}{\sqrt{2\pi }}\mathrm{exp}\left[\frac{-{\left[{y}_{\mathrm{min}}-\widehat{y}\left({\varvec{x}}\right)\right]}^{2}}{2{\widehat{s}}^{2}}\right],$$

(16)

where $\mathrm{erf}$ indicates an error function,${y}_{\mathrm{min}}$ is the current minimum while $\widehat{s}$ is standard deviation obtained from the constructed surrogate model, which is obtained by:

$${\widehat{s}}^{2}={\sigma }^{2}\left[1-{{\varvec{\psi}}}^{T}{\Psi }^{-1}{\varvec{\psi}}+\frac{{\left[1-{1}^{T}{\Psi }^{-1}{\varvec{y}}\right]}^{2}}{{1}^{T}{\Psi }^{-1}1}\right].$$

(17)

All the quantities used in Eqs. (16) and (17) are obtained during the construction of the surrogate model. In the Bayesian optimization, $EI\left({\varvec{x}}\right)$ is used to locate an additional sample point, i.e., a candidate of experimental conditions. The candidate of parameters ${\varvec{x}}$ is searched also by a genetic algorithm using $EI\left({\varvec{x}}\right)$ as a cost function.

Constraints associated with monitored quantities such as surface roughness can be considered in the Bayesian optimization procedure by considering $EI\left({\varvec{x}}\right)PF({\varvec{x}})$ with $PF\left({\varvec{x}}\right)$ represented by:

$$PF({\varvec{x}})=\frac{1}{\widehat{s}\sqrt{2\pi }}{\int }_{0}^{\infty }\mathrm{exp}\left[\frac{-{\left[G\left(x\right)-{g}_{\mathrm{min}}-\widehat{g}\left({\varvec{x}}\right)\right]}^{2}}{2{\widehat{s}}^{2}}\right]dG,$$

(18)

where $g$ is the constraint function, ${g}_{\mathrm{min}}$ is the constraint value, $G\left(x\right)$ is a random variable. In the present case, surface roughness $Ra$ is considered as a constraint.